Properties

Label 2-1950-65.64-c1-0-33
Degree $2$
Conductor $1950$
Sign $0.447 + 0.894i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s i·3-s + 4-s i·6-s + 2.60·7-s + 8-s − 9-s i·12-s − 3.60i·13-s + 2.60·14-s + 16-s + 2.60i·17-s − 18-s − 2.60i·19-s − 2.60i·21-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 0.984·7-s + 0.353·8-s − 0.333·9-s − 0.288i·12-s − 0.999i·13-s + 0.696·14-s + 0.250·16-s + 0.631i·17-s − 0.235·18-s − 0.597i·19-s − 0.568i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.043517170\)
\(L(\frac12)\) \(\approx\) \(3.043517170\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + 3.60iT \)
good7 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2.60iT - 17T^{2} \)
19 \( 1 + 2.60iT - 19T^{2} \)
23 \( 1 + 8.60iT - 23T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 - 5.21T + 37T^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 5.21T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 5.21iT - 59T^{2} \)
61 \( 1 - 3.21T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 5.21iT - 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 17.2T + 83T^{2} \)
89 \( 1 + 0.788iT - 89T^{2} \)
97 \( 1 - 8.60T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.668348362243521394352442805755, −8.251903583232766198974430732702, −7.41210413525799981451241495524, −6.62177325833450450079543937766, −5.81006821302598130586564265338, −4.98948607954711413229549173281, −4.25658310691202546604033946740, −3.01997949324377170511304367321, −2.17615426115419090488718904969, −0.937642589169117203571347645140, 1.47251143750536346575903126275, 2.55908265497268308462178652452, 3.75076982524780745312114159860, 4.39722650160371549626128221363, 5.20376082169551133218427164100, 5.87838374082138909008331812154, 6.92977136893596960614438086066, 7.72893750638819803229236175753, 8.469294065498406970972393510886, 9.502428570152138860497185761106

Graph of the $Z$-function along the critical line